Global continuum and multiple positive solutions to a \(p\)-Laplacian boundary-value problem. (English) Zbl 1260.34045

Kim, Chan-Gyun; Shi, Junping

Electron. J. Differ. Equ. 2012, Paper No. 106, 12 p. (2012).

Summary: A \(p\)-Laplacian boundary-value problem with positive nonlinearity is considered. The existence of a continuum of positive solutions emanating from \((\lambda,u)=(0,0)\) is shown, and it can be extended to \(\lambda=\infty\). Under an additional condition on the nonlinearity, it is shown that the positive solution is unique for any \(\lambda>0\); thus the continuum \(\mathcal C\) is indeed a continuous curve globally defined for all \(\lambda>0\). In addition, by the upper and lower solutions method, existence of three positive solutions is established under some conditions on the nonlinearity.

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C23	Bifurcation theory for ordinary differential equations

Keywords:

upper and lower solution; positive solution; \(p\)-Laplacian; uniqueness; multiplicity